| L(s) = 1 | + 2-s + 3-s + 4-s − 0.406·5-s + 6-s + 8-s − 2·9-s − 0.406·10-s + 0.460·11-s + 12-s − 2.18·13-s − 0.406·15-s + 16-s − 3.51·17-s − 2·18-s + 2.32·19-s − 0.406·20-s + 0.460·22-s − 9.38·23-s + 24-s − 4.83·25-s − 2.18·26-s − 5·27-s − 7.32·29-s − 0.406·30-s + 7.92·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.181·5-s + 0.408·6-s + 0.353·8-s − 0.666·9-s − 0.128·10-s + 0.138·11-s + 0.288·12-s − 0.606·13-s − 0.104·15-s + 0.250·16-s − 0.852·17-s − 0.471·18-s + 0.533·19-s − 0.0908·20-s + 0.0981·22-s − 1.95·23-s + 0.204·24-s − 0.966·25-s − 0.428·26-s − 0.962·27-s − 1.36·29-s − 0.0742·30-s + 1.42·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 0.406T + 5T^{2} \) |
| 11 | \( 1 - 0.460T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 + 9.38T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 43 | \( 1 - 6.02T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 8.86T + 67T^{2} \) |
| 71 | \( 1 - 0.273T + 71T^{2} \) |
| 73 | \( 1 - 9.70T + 73T^{2} \) |
| 79 | \( 1 + 0.0468T + 79T^{2} \) |
| 83 | \( 1 + 6.78T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.940606458856640758467490240359, −7.49570414740556329601907104733, −6.44158520508717610739474649454, −5.87080647040453957506656282476, −5.02278821259031653789098064411, −4.12378933935812907950622329718, −3.51094065000205308736237477378, −2.53470578759653124826983685835, −1.87262484621551430448787001945, 0,
1.87262484621551430448787001945, 2.53470578759653124826983685835, 3.51094065000205308736237477378, 4.12378933935812907950622329718, 5.02278821259031653789098064411, 5.87080647040453957506656282476, 6.44158520508717610739474649454, 7.49570414740556329601907104733, 7.940606458856640758467490240359
